Skip to main content
Log in

Calculus on Straight Singular Cones

  • Original Research Article
  • Published:
La Matematica Aims and scope Submit manuscript

Abstract

The analysis on singular spaces requires a systematic calculus of pseudo-differential operators on infinite cones with compact singular link. This will be developed in the present article for links with edge.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Data Availability

Not applicable.

References

  1. Chang, D.-C., Schulze, B.-W.: Ellipticity on spaces with higher singularities. Sci. China Math. 60(11), 2053–2076 (2017). https://doi.org/10.1007/s11425-016-0519-9

    Article  MathSciNet  MATH  Google Scholar 

  2. Chang, D.-C., Khalil, S., Schulze, B.-W.: Corner spaces with symbol hierarchies. Adv. Appl. Clifford Algebras 31(4), 31–47 (2021). https://doi.org/10.1007/s00006-021-01130-x

    Article  MATH  Google Scholar 

  3. Boundary problems for pseudo-differential operators: Boutet de Monvel, L. Acta Math. 126, 11–51 (1971)

    MathSciNet  Google Scholar 

  4. Cordes, H.O.: A global parametrix for pseudo-differential operators over \({\mathbb{R}}^{n}\) with applications. Reprint SFB 72. Uni. Bonn (1976)

  5. Dorschfeldt, Ch.: Algebras of Pseududifferential Operators Near Edge and Corner Singularities, Mathematical Research, vol. 102. Wiley-VCH, Weinheim (1998)

    MATH  Google Scholar 

  6. Eskin, G.I.: Boundary Value Problems for Elliptic Pseudodifferential Equations. Mathematical Monographs, vol. 52. American Mathematical Society, Providence (1980). Transl. of Nauka, Moskva (1973)

  7. Gil, J.B., Schulze, B.-W., Seiler, J.: Holomorphic operator-valued symbols for edge-degenerate pseudo-differential operators. In: Differential Equations, Asymptotic Analysis, and Mathematical Physics. Math. Research, vol. 100, pp. 113–137. Akademie Verlag, Berlin (1997)

  8. Gil, J.B., Schulze, B.-W., Seiler, J.: Cone pseudodifferential operators in the edge symbolic calculus. Osaka J. Math. 37, 221–260 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Hörmander, L.: Pseudo-differential operators and non-elliptic boundary problems. Ann. Math. 83(1), 129–200 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kapanadze, D., Schulze, B.-W.: Crack Theory and Edge Singularities. Kluwer, Dordrecht (2003)

    Book  MATH  Google Scholar 

  11. Rabinovič, V.S.: Pseudodifferential operators in non-bounded domains with conical structure at infinity. Mat. Sb. 80, 77–97 (1969)

    MathSciNet  Google Scholar 

  12. Rempel, S., Schulze, B.-W.: Asymptotics for Elliptic Mixed Boundary Problems (Pseudo-differential and Mellin Operators in Spaces with Conormal Singularity). Math. Res., vol. 50. Akademie-Verlag, Berlin (1989)

  13. Schrohe, E.: Spaces of weighted symbols and weighted Sobolev spaces on manifolds. In: Cordes, H.-O., Widom, H., Gramsch, B. (eds.) Pseudo-Differential Operators. Lecture Notes of Mathematics, vol. 1256, pp. 360–377. Springer, Berlin (1986)

    Chapter  MATH  Google Scholar 

  14. Shubin, M.A.: Pseudodifferential operators in \({\mathbb{R} }^{n}\). Dokl. Akad. Nauk SSSR 196, 316–319 (1971)

    MathSciNet  Google Scholar 

  15. Harutyunyan, G., Schulze, B.-W.: Elliptic Mixed. Transmission and Singular Crack Problems. European Mathematical Society, Zürich (2008)

    MATH  Google Scholar 

  16. Hwang, I.L.: The \(L^2\) boundedness of pseudo-differential operators. Trans. Am. Math. Soc. 302, 55–76 (1987)

    Google Scholar 

  17. Jachow, H.: Locally Convex Spaces. Teubner, Stuttgart (1981)

    Book  Google Scholar 

  18. Seiler, J.: Continuity of edge and corner pseudo-differential operators. Math. Nachr. 205, 163–182 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Seiler, J.: Pseudodifferential calculus on manifolds with non-compact edges. Ph.D. thesis, University of Potsdam (1997)

  20. Seiler, J.: Mellin and Green pseudodifferential operators associated with non-compact edges. Integr. Equ. Oper. Theory 31, 214–245 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Schulze, B.-W.: Pseudo-Differential Operators on Manifolds with Singularities. North-Holland, Amsterdam (1991)

    MATH  Google Scholar 

  22. Krainer, T.: A calculus of abstract edge pseudodifferential operators of type \(\rho , \delta \). In: Escher, J., et al. (eds.) Elliptic and Parabolic Equations. Springer Proceedings in Mathematics and Statistics, pp. 179–207. Springer, Hannover (2013)

  23. Schulze, B.-W.: Boundary Value Problems and Singular Pseudo-Differential Operators. Wiley, Chichester (1998)

    MATH  Google Scholar 

  24. Hedayat Mahmoudi, M., Schulze, B.-W., Tepoyan, L.: Continuous and variable branching asymptotics. J. Pseudo-Differ. Oper. Appl. 6(1), 69–112 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to M. Hedayat Mahmoudi or B.-W. Schulze.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author D.-C. Chang is partially supported by a McDevitt Endowment Fund at Georgetown University.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chang, DC., Hedayat Mahmoudi, M. & Schulze, BW. Calculus on Straight Singular Cones. La Matematica 2, 616–634 (2023). https://doi.org/10.1007/s44007-023-00057-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s44007-023-00057-y

Keywords

Mathematics Subject Classification

Navigation